Hicksian demand function

As Hicksian demand function (also: compensated demand function ) is called in the micro-economic theory and in particular in the household theory, a function that specifies the demand for goods depending on their price and a certain ( minimum ) benefit level to be a total attained.

The demand function bears her name in reference to John Richard Hicks, who formalized the concept of compensated demand in 1939 for the first time.

Definition and meaning

Formal representation

It is initially based on an output minimization problem by

Is given, in which continuous, strictly increasing, differentiable and strictly quasikonkav was. is the vector of quantities of goods demanded and the corresponding price vector.

In the above problem, the total expenditure is minimized for the goods from the shopping cart, but a certain level of utility is to be maintained. The solution of such an output minimization problem is intended a function that indicates what amount should be requested from the respective goods in order to achieve the given level of utility as inexpensively as possible. It is therefore a function of the price vector and the fixed utility levels.

This is known as Hicksian demand as given and agreed upon.

Simplified representation of the two-goods case

The output problem reduces to the classical two-goods case

Thus minimized the total expenditure for the two goods and with their respective prices or. The solution of the minimization problem are two functions and that a function of the prices of goods ( all goods! ) And the least desired benefit level show how much optimally from Good 1 and Good 2 is to be consumed. These functions are called Hicksian demands and writes or.

Example

In the example, that the price of good 1 and that of Good second The consumer relates its benefits exclusively from these two estates. His utility function is. We formulate the following optimization problem now for simplicity equality constraint ( ), which is justified by the properties of the utility function. The minimization problem is:

The corresponding Lagrangian reads. Are the optimality conditions

Of (1) and (2) follows and used in (3), finally followed by the requests Hicksian

Note that the Hicksian demands for the two goods are identical, if the price of good 2 just twice as high as that of good 1.

Properties of the Hicksian demand function

It can be shown that under the given conditions, among others, has the following properties:

• Homogeneity of degree zero in.
• Convex set. is a convex set.
• Monotonically decreasing in own price: The derivative of the Hicksian demand for a commodity the price of this good, is not positive:

Related to Marshallian demand

Although Hicksian demand functions play an important role in many areas of budget theory, they are not in itself directly observable and thus a hypothetical construct. While Marshallian demand functions of an empirical analysis are in principle accessible - you can see how a person's demand changes for a good, if changed their income or the price of goods, for example - this does not apply to compensated demand functions, as its core element, the benefit assessment hidden from the outside remains a consideration. However, there Hicks'scher between demand and its Marshallian counterpart a close relationship, which allows, for example, the derivative of the Hicksian demand for a commodity under its own or a different price - that is - based on partial derivatives of the Marshallian demand function in calculate ( Slutsky decomposition ).

In fact, Marshallian and Hicksian demand functions are also even even functionally connected:

Duality of marshall shear and Hicks'scher demand function: Let the preference ordering of consumers representable by a real-valued and continuous, strictly increasing and strictly quasikonkave utility function and represents. Be on the Marshallian demand for a commodity, an output function and the indirect utility function for income level. Then: